Garrett Parzych

Garrett Parzych, Joshua J. Daymude

We introduce the problem of adaptive self-organization in which the nodes of an anonymous, synchronous dynamic network must distributively change the collective distribution of their responses (or "colors") as a function of time-varying environmental signals, even when these signals are only perceived locally and the network topology changes adversarially. Specifically, a signal adversary may change the type of signal and which node(s) witness that signal arbitrarily between rounds. If a signal (or lack thereof) $s$ persists in the system for sufficiently long, the dynamic network must stabilize such that nodes' colors reach and remain in a distribution closely approximating $r(s)$, a goal distribution defined by the problem instance. We first prove that if nodes are deterministic, the only solvable instances of adaptive-self organization are those with homogeneous goal distributions, i.e., those where all nodes must stabilize with the same color. We then present a linear-time, logarithmic-memory, deterministic algorithm for this subclass of instances that works even when the multiplicity and location of signal witnesses change arbitrarily. When nodes know $n$, the number of nodes in the network, a small adaptation of this algorithm achieves a stronger convergence property in which adversarial edge and signal dynamics are entirely unable to disturb stabilized configurations. Finally, we present a randomized extension of these algorithms that solves arbitrary (i.e., not necessarily homogeneous) instances of adaptive self-organization with high probability when nodes know the goal distributions.

Garrett Parzych, Joshua J. Daymude

Broadcast is a ubiquitous distributed computing problem that underpins many other system tasks. In static, connected networks, it was recently shown that broadcast is solvable without any node memory and only constant-size messages in worst-case asymptotically optimal time (Hussak and Trehan, PODC'19/STACS'20/DC'23). In the dynamic setting of adversarial topology changes, however, existing algorithms rely on identifiers, port labels, or polynomial memory to solve broadcast and compute functions over node inputs. We investigate space-efficient, terminating broadcast algorithms for anonymous, synchronous, 1-interval connected dynamic networks and introduce the first memory lower bounds in this setting. Specifically, we prove that broadcast with termination detection is impossible for idle-start algorithms (where only the broadcaster can initially send messages) and otherwise requires $Ω(\log n)$ memory per node, where $n$ is the number of nodes in the network. Even if the termination condition is relaxed to stabilizing termination (eventually no additional messages are sent), we show that any idle-start algorithm must use $ω(1)$ memory per node, separating the static and dynamic settings for anonymous broadcast. This lower bound is not far from optimal, as we present an algorithm that solves broadcast with stabilizing termination using $\mathcal{O}(\log n)$ memory per node in worst-case asymptotically optimal time. In sum, these results reveal the necessity of non-constant memory for nontrivial terminating computation in anonymous dynamic networks.